Mr. Bloomberg, after all, became a billionaire by founding a company that was built on mining statistics.

And since becoming mayor nearly a decade ago, he has minutely quantified virtually every detail of his government, from the number of mentally ill inmates in city jails to the days left in his current term. A video screen at City Hall regularly updates the progress of dozens of agencies in meeting their goals.

But for a mayor who advertised his managerial expertise to win a third term, some of his administration’s own numbers show a clear slump in the performance of agencies Mr. Bloomberg oversees as the city’s chief executive.

While many performance indicators registered gains since 2001, a higher share declined and fewer improved during most of the last fiscal year than in any year since Mr. Bloomberg’s first term. The slump, magnified in the past two years, has apparently not been lost on his constituents. Before the mayor’s timely, aggressive, reassuring and highly visible response to Tropical Storm Irene — in contrast to his administration’s lackadaisical handling of last December’s blizzard — his job approval rating had declined to its lowest point since 2005, according to a recent New York Times/CBS News poll.

“Clearly you get a sense of a city straining, certainly in some areas,” said Doug Turetsky, chief of staff of the city’s Independent Budget Office. “You can see families staying longer in shelters. Child welfare workers’ caseloads are creeping up. It certainly gives pause.”

For most of the fiscal year that ended June 30, according to the Citywide Performance Reporting system initiated by the mayor, responses to emergencies by the Police, Fire and Buildings Departments were slower than in the year before. More water mains broke, the streets were dirtier, the backlog of broken hydrants increased and the time it took to arrange for a home health attendant doubled to 26 days, according to the administration’s report card on how city services were delivered. Medical examiners took a week longer to sift DNA evidence in sexual assault cases. The average time between a passenger’s request for a taxi complaint hearing and a final decision rose to 63 days from 56 the year before.

Among seven broad services provided by the city, a majority of indicators improved in only a single category: economic development. In two categories, public safety and education, more indicators declined than improved.

Among all of the 478 indicators or measurements of progress monitored by the mayor’s office, about 48 percent improved and 45 percent declined, according to an analysis by The New York Times — a record that may undercut Mr. Bloomberg’s reputation for applying business skills to city government even during a recession. Administration officials defended their performance, saying the sagging economy inevitably led to some service cuts, even as agencies are still expected to show year-to-year improvements.

“It’s very important to remember the premise here,” said Stu Loeser, the mayor’s chief spokesman, referring to the online color-coded Citywide Performance Rating. “The dashboard is designed with the expectation that we do better than the previous year — if agencies don’t do as well as before, they get a yellow or red. Green only goes for significant improvements — hard to do. It’s a high standard we set for ourselves.”

Henry J. Stern, a former parks commissioner who is the director of New York Civic, a nonpartisan government watchdog group, said the decline in services was self-evident. “The snow wasn’t picked up,” he said, referring to the city’s slow response to the blizzard in December. “People know that; you don’t need statistics.” Some of the sheen of Mr. Bloomberg’s business acumen has also faded as a result of the corruption charges and investigation involving CityTime, the administration’s automated payroll project and another data-driven effort embraced by the mayor.

“Given the magnitude of the CityTime scandal, I think the mayor has already squandered most of his smart management reputation,” said James A. Parrott, chief economist for the Fiscal Policy Institute, a labor-supported research and advocacy group.

Since 2001, when Mr. Bloomberg was first elected, major felony crimes have declined by 39 percent (though murders have increased since 2009) and traffic fatalities have declined by nearly 40 percent, according to a 10-year snapshot of indicators chosen by the mayor’s office.

The response time for crimes in progress is 8 minutes 54 seconds, better than in 2001, when it was 10 minutes 6 seconds, but is climbing again. So is the response time to medical emergencies (a full minute higher than in 2001 after declining). Street cleanliness is higher, but slipping again. The number of children in foster care, civilian fire fatalities and the infant mortality rate have all plunged and the number of families placed in permanent housing has increased.

More potholes have been repaired (305,001 in 2011 compared with 121,331 in 2001), and response times to complaints about clogged sewers have improved.

“While overall performance is important, we think New Yorkers care most about key quality of life measures like crime and cleanliness, which have dramatically improved over the past 10 years,” said Julie Wood, a mayoral spokeswoman.

Elizabeth Weinstein, director of the Mayor’s Office of Operations, said that the raw accounting of declining and improving performance could be misleading. For example, the number of building-related fatalities has declined, but is that because of better safety monitoring or because the economy has put a crimp in construction? That senior centers are serving fewer lunches is considered a decline, but maybe it means fewer elderly people are hungry or more are relying on food stamps or other substitutes.

Moreover, managing a city is not necessarily the same as running a business, where improved productivity might generate more sales and revenue that could be invested to improve results further.

“If they have one fabulous February, then this year you’re going to be yellow or red unless you match it,” Ms. Weinstein said, referring to the color codes the city uses to grade agencies.

Some analysts gave Mr. Bloomberg credit for being transparent about his administration’s performance, be it good or bad, especially compared with some of his City Hall predecessors.

“I was surprised when I saw the amount that was declining,” said Mr. Turetsky of the Independent Budget Office, “but you could actually give the Bloomberg administration a high mark for honesty when you think back to the Giuliani era, when the sun was always shining.”

A version of this article appears in print on August 30, 2011, on Page A18 of the New York edition with the headline: Statistics, Beloved by Mayor, Show a Slump in City Services. Order Reprints|Today's Paper|Subscribe